Never seen dropouts like these before.
 

It's a turn of the century Remington. Looks like a New Departure Model A hub. I wonder if there's some sort of cam arrangement that fits these for chain tensioning.
http://cgi.ebay.com/ws/eBayISAPI.dll...t_13229wt_1110

Those aren't dropouts. They're called loopdrops. Er, droploups. Er, doodlyoop by diggity ring things!

Woah, those dropout are weird...

Looks like a New Departure hub sitting on the ground but the laced to the wheel looks like the original. Might be worth searching the Remington patents for "hub" to see what comes up. The Google patent search works amazingly well by the way.

I have an old rod-brake bike with a seatpost like the one on the eBay bike. What is that style called and how does a saddle attach?

I've seen it here on the forums before, but I couldn't begin to tell you where. If you want to see some different idea's about how a bike should be built then check this site out...

http://www.oldbike.eu/museum/

Most folks just call it a "lucky 7". You rotate the seat clamp from vertical to horizontal and clamp tight. Gotta have the right size clamp, though.

I recognize it too.

1890's?
NICE!
But I can't find parts for what I have much-less that old!
Nice Loopdrops!